Operations with complex numbers represented in algebraic form

Arithmetic operations with complex numbers are similar to operations with algebraic binomials.

Addition of complex numbers

To add two complex numbers, you need to add the real numbers of their real and imaginary parts. The formula for adding two complex numbers a+bi and c+di is written as follows:

(a + bi) + (c + di) = a+c+(b+d)i.

Let us give the addition formula for two complex numbers (a;b) and (c;d), represented as ordered pairs of real numbers:

(a;b) + (c;d) = (a+c;b+d).

The sum of two complex conjugate numbers is always equal to a real number:

(a + bi) + (a - bi) = 2a.

Examples.
1) (3+4i) + (2-4i) = 5;
2) (-2+i) + (1+3i) = -1+4i;
3) (-5+5i) + (5-3i) = 2i;
4) 6 + (-3-8i) = 3-8i.

Subtraction of complex numbers

Subtraction of complex numbers is reduced to subtraction of real numbers of their real and imaginary parts:

(a + bi) - (c + di) = a-c+(b-d)i.

For two complex numbers (a;b) and (c;d), represented as ordered pairs of real numbers, a similar formula is written as follows:

(a;b) - (c;d) = (a-c;b-d).

Examples.
1) (3+4i) - (2-4i) = 1+8i;
2) (-2+i) - (1+3i) = -3-2i;
3) (-5+5i) - (5-3i) = -10+8i;
4) 6 - (-3-8i) = 9+8i.

Multiplication of complex numbers

Multiplication of complex numbers is carried out in the same way as multiplication of algebraic binomials. Let us present the derivation of the corresponding formula, taking into account that i²= -1:

(a + bi)(c + di) = ac + adi + bci + bdi² = ac-bd + (ad+bc)i.

For two complex numbers (a;b) and (c;d), represented as ordered pairs of real numbers, a similar formula is written as follows:

(a;b)(c;d) = (ac-bd;ad+bc).

The product of two conjugate complex numbers a+bi and a-bi is always a real number, and non-negative:

(a + bi)(a - bi) = a² - abi + abi - b²i² = a²+b².

Examples.
1) (3+4i)(2-4i) = 6-12i+8i+16 = 22-4i;
2) (-2+i)(1+3i) = -2-6i+i-3 = -5-5i;
3) (-5+5i)(5-3i) = -25+15i+25i+15 = -10+40i;
4) 6(-3-8i) = -18-48i.
5) (3-2i)(3+2i) = 9+4 = 13.

Division of complex numbers

Dividing a complex number z₁ = a+bi by a complex number z₂ = c+di s done by multiplying the numerator and denominator of the fraction z₁/z₂ by the complex conjugate of the denominator. As a result, the denominator is a real positive number, since z₂≠0. Thus, finding the quotient z₁/z₂ is reduced to multiplying z₁ by the complex conjugate of the denominator and dividing the resulting product by a positive number. We get the formula for the quotient of division z₁/z₂:

z₁/z₂

=

a+bi/c+di

=

(a+bi)(c-di)/(c+di)(c-di)

=

ac+bd/c²+d²

+

bc-ad/c²+d²

i.

A similar formula for two complex numbers (a;b) and (c;d), represented as ordered pairs of real numbers, is written as follows:

(a;b)/(c;d)

= (

ac+bd/c²+d²

;

bc-ad/c²+d²

)

.

The formulas given are too cumbersome and difficult to remember. Therefore, for dividing complex numbers, it is recommended to use the formula
Examples.
1)

Raising complex numbers to a power with an integer exponent

Raising complex numbers to a power with an integer exponent is done according to the same formulas and rules as raising real numbers to a power. For any complex number z≠0 and integer m, n

z⁰ = 1;   z^-n = (1/z)ⁿ;   zⁿz^m=z^n+m;   zⁿ:z^m=z^n-m.

Examples.
1) (3+2i)² = 9+12i-4 = 5+12i;
2) (2-5i)³ = 2³-3*2²*(5i)+3*2*(5i)²-(5i)³ = 8-60i-150+125i = -142+85i;
3) (1+i)⁴ = (1+i)²(1+i)² = (1+2i-1)(1+2i-1) = 2i*2i = -4;
4) (1+i)^-2 = 1/(1+i)² = 1/(1+2i-1) = 1*(-2i)/(2i*(-2i)) = -2i/4 = -(1/2)i.